Saturday, August 30, 2014

Gillies’ Philosophical Theories of Probability, Chapter 2

As Gillies notes, the “Classical” interpretation was the earliest theory of probability and its most important statement was by Pierre-Simon Laplace (1749–1827) in his Essai Philosophique sur les Probabilités [A Philosophical Essay on Probabilities] (1814). However, it is largely of historical interest now, and has no supporters today (Gillies 2000: 3).

Laplace’s Essai Philosophique sur les Probabilités (1814) made the assumption of universal determinism on the basis of Newtonian mechanics (Gillies 2000: 16). Laplace argued that an agent with perfect knowledge of Newtonian mechanics and all matter could predict the future state of the universe. It is only human ignorance that prevents perfect forecasting, and leads us to calculate probabilities (Gillies 2000: 17). Thus probability, according to Laplace, is a measure of human ignorance (Gillies 2000: 21).

Laplace’s formula for calculating probabilities is the familiar one where the probability P(E) of any event E in a finite sample space S, where all outcomes are equally likely, is the number of outcomes for E divided by the total number of outcomes in S.

But there is an obvious limitation with this, as pointed out by the later advocates of the frequency theory of probability like Richard von Mises: what if our outcomes are not equiprobable? (Gillies 2000: 18). Thus the Classical interpretation of probability has a serious shortcoming.

As probability theory came to be increasingly applied to phenomena in the natural and social sciences in the 19th century, its limiting assumption of equiprobable outcomes was exposed as a problem, and the relative frequency approach was developed as a new and alternative theory.